My answer would be "No, but..."
No, the "using your attack action to grapple/shove instead" rules would not apply, since it's not a creature - you quoted the relevant rules already.
However, the list of actions one can take in combat is non-exhaustive, as per "Actions in Combat" in the SRD:
"When you describe an action not detailed elsewhere in the rules, the GM tells you whether that action is possible and what kind of roll you need to make, if any, to determine success or failure."
Far be it from me to discourage the attempt - the player can absolutely try to grab onto something conjured by a spell.
From there, we'd have to decide what happens next. I'd rule as follows, based on each spell's description:
Bigby's Hand is a relatively simple decision. It's a visible, solid object with its own Strength score, so an attempt to move or hold it in place would be a contested Strength check between the person trying to move/grab it and the Hand - same as a Grapple/Shove attempt against a creature.
Guardian of Faith, there's nothing there to grab onto or push - it doesn't have an AC, hit points, or any physical ability scores, just an area around itself where Radiant damage happens. You could try to grab or push it, but it'd have no more effect than trying to grab onto a Silent Image.
Spiritual Weapon doesn't have its own stats, but it deals Force damage which indicates it's tangible - so I as a DM would rule it's a contested check of Strength versus the caster's Spellcasting Ability Modifier, given the attacks it makes are also based on the caster's Spell Attack Modifier.
Unseen Servant is an invisible, shapeless force - but it does have an AC, a hit point, and a Strength score. You can try to grab onto it or shove it directly, but given it's invisible and amorphous, it'd be harder than its paltry Strength indicates on account of being hard to actually grab hold of. Probably a Strength check at disadvantage, versus a flat roll on the part of the Servant (AC10 = ±0 Dex modifier).
Ripping an object from its grasp would be much easier, given neither its invisibility nor its formlessness would help it there - straight contested Strength check, it has the stat for a reason.